Problem: Determine how many solutions exist for the system of equations. ${-18x-3y = 21}$ ${y = -7-6x}$
Answer: Convert both equations to slope-intercept form: ${-18x-3y = 21}$ $-18x{+18x} - 3y = 21{+18x}$ $-3y = 21+18x$ $y = -7-6x$ ${y = -6x-7}$ ${y = -7-6x}$ ${y = -6x-7}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -6x-7}$ ${y = -6x-7}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-18x-3y = 21}$ is also a solution of ${y = -7-6x}$, there are infinitely many solutions.